Before giving a proof, we will need the following Lemma where we describe a rigidification of each stratum \(J_{(G,D)}\)  of \(\overline{\mathcal {J}}_{g,n}\) . For a similar description of the strata of \(\overline{\mathcal {M}}_{g,n}\)  (and also of their closure), see [1]} (or [2]} for a more algebraic proof).
